Abstract

When allowance is made for the instability of the excited states of hydrogen it is necessary to replace the equation of Salpeter & Bethe (1951) by a set of coupled integral equations for representatives of the state vector. These representatives correspond to an electron-proton bound state and also to the electron and proton with any number of photons present. The coupled equations can be reduced to a single integral equation, which gives the electronproton bound state as an eigenstate of a modified propagator. The modified propagator is related to the two-body propagator of Salpeter & Bethe. The difference between the first approximation to the modified propagator and the first approximation to the two-body compound propagator (Eden 1952, 1953) can be represented by a displacement of its singularity in total energy-momentum space. This displacement gives in a relativistic form all the relevant contributions to the Lamb shift to this order; these include the contribution from low-energy transverse photons crossing over an arbitrary number of longitudinal photons; previously this term has always been deduced by physical arguments and obtained by non-relativistic methods (Bethe 1947; Salpeter 1952). The displacement of the singularity also gives decay coefficients to this order in the charge. The method can readily be extended to higher approximations.

Footnotes

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